An ensemble Kalman filter implementation based on shrinkage estimators of the precision matrix via modified Cholesky decomposition

We introduce a family of shrinkage estimators of the background‐error precision matrix for the ensemble Kalman filter (EnKF). In contrast with classical Ledoit–Wolf shrinkage, which acts on the covariance and uses a scaled‐identity target, the proposed estimators write the inverse background‐error covariance as a convex combination of two data‐driven precision estimators and select the mixing weight under three principled criteria. Working in precision space is natural because, for the Gaussian–Markov random fields that describe geophysical error structure, it is the precision and not the covariance that is approximately sparse, and because the analysis operator then inherits that sparsity. Two of the three estimators mix a sparse modified‐Cholesky precision, obtained from local conditional regressions, with the truncated singular‐value decomposition (SVD) pseudo‐inverse of the sample covariance; one weight is available in closed form under Stein's loss, and the other follows from a one‐dimensional convex optimization of the posterior‐covariance trace. The third instead mixes two modified‐Cholesky precisions at different radii of influence, yielding a multi‐scale estimator that is everywhere well‐conditioned, with a weight that requires no resampling. All three weights are computable without reference to the unknown precision, and the analysis update uses sparse and matrix‐free operations, so that storage scales linearly with the model dimension. Numerical experiments with the Lorenz‐96 model and with the Simplified Parametrizations, primitivE‐Equation DYnamics (SPEEDY) atmospheric general circulation model compare the three estimators against the modified‐Cholesky EnKF and the local ensemble transform Kalman filter (LETKF). All filters perform comparably when observations are dense. As the network is thinned, the multi‐scale estimator attains an analysis error an order of magnitude below the modified‐Cholesky baseline for Lorenz‐96, where LETKF loses stability altogether at the localization radius that had been optimal under full observation, while for SPEEDY the Stein weight gives the lowest errors in the wind components and in lower‐tropospheric humidity.